x^4-32x^2+32=0

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Solution for x^4-32x^2+32=0 equation:


Simplifying
x4 + -32x2 + 32 = 0

Reorder the terms:
32 + -32x2 + x4 = 0

Solving
32 + -32x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + -32x2 + -32 + x4 = 0 + -32

Reorder the terms:
32 + -32 + -32x2 + x4 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + -32x2 + x4 = 0 + -32
-32x2 + x4 = 0 + -32

Combine like terms: 0 + -32 = -32
-32x2 + x4 = -32

The x term is -32x2.  Take half its coefficient (-16).
Square it (256) and add it to both sides.

Add '256' to each side of the equation.
-32x2 + 256 + x4 = -32 + 256

Reorder the terms:
256 + -32x2 + x4 = -32 + 256

Combine like terms: -32 + 256 = 224
256 + -32x2 + x4 = 224

Factor a perfect square on the left side:
(x2 + -16)(x2 + -16) = 224

Calculate the square root of the right side: 14.966629547

Break this problem into two subproblems by setting 
(x2 + -16) equal to 14.966629547 and -14.966629547.

Subproblem 1

x2 + -16 = 14.966629547 Simplifying x2 + -16 = 14.966629547 Reorder the terms: -16 + x2 = 14.966629547 Solving -16 + x2 = 14.966629547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = 14.966629547 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = 14.966629547 + 16 x2 = 14.966629547 + 16 Combine like terms: 14.966629547 + 16 = 30.966629547 x2 = 30.966629547 Simplifying x2 = 30.966629547 Take the square root of each side: x = {-5.564766801, 5.564766801}

Subproblem 2

x2 + -16 = -14.966629547 Simplifying x2 + -16 = -14.966629547 Reorder the terms: -16 + x2 = -14.966629547 Solving -16 + x2 = -14.966629547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + x2 = -14.966629547 + 16 Combine like terms: -16 + 16 = 0 0 + x2 = -14.966629547 + 16 x2 = -14.966629547 + 16 Combine like terms: -14.966629547 + 16 = 1.033370453 x2 = 1.033370453 Simplifying x2 = 1.033370453 Take the square root of each side: x = {-1.016548303, 1.016548303}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-5.564766801, 5.564766801, -1.016548303, 1.016548303}

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